Research Article Detailed Simulation of Complex Hydraulic Problems...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 928309 14 pageshttpdxdoiorg1011552013928309

Research ArticleDetailed Simulation of Complex Hydraulic Problems withMacroscopic and Mesoscopic Mathematical Methods

Chiara Biscarini1 Silvia Di Francesco2 Fernando Nardi1 and Piergiorgio Manciola3

1 Warredoc University for Foreigners of Perugia Via XIV Settembre 06100 Perugia Italy2 CIPLA Interuniversitary Centre for Environment University of Perugia Piazza Universita 1 06125 Perugia Italy3 DICA Department of Civil and Environmental Engineering University of Perugia Via G Duranti 93 06125 Perugia Italy

Correspondence should be addressed to Chiara Biscarini chiarabiscariniunistrapgit

Received 30 January 2013 Accepted 15 March 2013

Academic Editor Carlo Cattani

Copyright copy 2013 Chiara Biscarini et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The numerical simulation of fast-moving fronts originating from dam or levee breaches is a challenging task for small scaleengineering projects In this work the use of fully three-dimensional Navier-Stokes (NS) equations and lattice Boltzmann method(LBM) is proposed for testing the validity of respectively macroscopic and mesoscopic mathematical models Macroscopicsimulations are performed employing an open-source computational fluid dynamics (CFD) code that solves the NS combinedwith the volume of fluid (VOF) multiphase method to represent free-surface flows The mesoscopic model is a front-trackingexperimental variant of the LBM In the proposed LBM the air-gas interface is represented as a surface with zero thickness thathandles the passage of the density field from the light to the dense phase and vice versa A single set of LBM equations representsthe liquid phase while the free surface is characterized by an additional variable the liquid volume fraction Case studies showadvantages and disadvantages of the proposed LBM and NS with specific regard to the computational efficiency and accuracy indealing with the simulation of flows through complex geometries In particular the validation of themodel application is developedby simulating the flow propagating through a synthetic urban setting and comparing results with analytical and experimentallaboratory measurements

1 Introduction

The detailed three-dimensional (3D) simulation of free-surface flows is a challenging task for engineering projects [1]dealing with open flow hydraulics at both the large andsmall scale because of the computational burden and thelarge parameterization needed for representing the physicsgeometry and boundary conditions Nevertheless in recenttimes the always larger availability of remotely sensed dataproviding digital topographic and land cover informationand the decreasing impact of the computational burdenmakethe application of small-scale high-resolution 3D numericalmodels feasible and attractive for engineering studies Inparticular it is foreseen that 3D hydraulic simulations shallsupport not only fluid mechanics analyses for engineeringdesign at the small scale but also large-scale applications in

urban planning river restoration and flood mitigation andmanagement projects

There are several recent works in the literature thatreport the ability of the Navier-Stokes (NS) method using thetwo-dimensional (2D) shallow water (SW) equations [2] tosimulate river surface flows even if the SW hypothesis is notalways valid for the inertial effects and significant gradients[3ndash5] In fact several fully three-dimensional NS simulationsare available in the literature for representing not only dam-breaks [6 7] but also unconfined surface flows along straightriver segments bends and confluences [8] as well as impulsewaves and tsunamis [4 9] and water falls [10] among others

The lattice Boltzmann method (LBM) is an alternativenumerical fluid dynamics scheme based on Boltzmannrsquoskinetic equation [11] that represents the flow dynamics atthe macroscopic level by incorporating a microscopic kinetic

2 Mathematical Problems in Engineering

approach that also preserves the conservation law [12 13]LBM has been adopted for the characterization of flows inporous media [14] and for multiphase flows [15ndash17] amongothers [18 19] The LBM demonstrated to work efficientlyfor describing the complex physics of non ideal flows andcomplex geometries [20] There are also several works inves-tigating the performances of the LBM in simulating shallowfree-surface flows [21 22] but with no emphasis on their usefor practical engineering applications

In this work a fully 3D front-tracking formulation of theLBM that was originally presented [23] for foam dynamicsand that was revisited for free-surface flow applications [24ndash26] is implemented validated and compared to the standardNS approach for transient hydraulic flows including a syn-thetic case of flow propagation through an urban setting

The storage effect of urban areas as well as the hydrographseparation effect resulting in different water transmissionrates was investigated in several works [27] Experiments ofdam breaks were conducted [28] to investigate the effect ofa flood flow propagating to inundate a single building [29]Nevertheless the application and potential validation of LBMfor urban modeling of flash floods [30] has not been ade-quately developed yet

The novel aspect of the proposedwork is the investigationfor the potential of the LBM to simulate the flooding in highlyurbanized areas with specific regard to the detailed char-acterization of instantaneous discharge variations like theone caused by dam or dyke breaks or by intense rainfall offlash floods producing multidimensional street surface flowpaths which is one of the most complex and difficulthazardous phenomena to manage prevent and control Theinteraction of water flow with anthropogenic features pro-duces complex 3D flow structures that shall be simulated bytaking into consideration all the resulting flow physics suchas hydraulics jumps flow constrictions and back pressureeffects Disregarding those geometric and hydraulic featureswould result in a misinterpretation of the estimated waterlevels providing an inaccurate flood hazard and risk charac-terization

In this study given the impossibility of gathering mea-sured data on such phenomena a comparative analysis of theperformance of LBM and NS in simulating flash flood prop-agation in urban areas is investigated by a comparison withexperimental data of dam break hydraulic effects on asynthetic case study represented by an ideal city

The present document is organized as follows in the nextsection the theoretical and numerical specifications of bothLBM and NS are introduced and described Then in the casestudy section the results of the application of the proposedmethods are insertedwith specific regard to (1) the test case ofthe LBM results compared to an analytical solution (2) theanalysis and comparison of LBMandNS codes applied on thesimulation of a rapidly varying flow along a simple geometryof a straight river reach (3) the evaluation and validation ofthe LBM and NS code application on the flow propagatingthrough a synthetic urban domain by means of comparisonwith experimental data of a physical model The last twosections represent respectively the discussion and conclusiveremarks

2 Methods

21 The Navier-Stokes Numerical Scheme The most generalmacroscopic model for hydraulic flows may be representedby the incompressible Navier-Stokes equations as follows

120588(120597u120597119905minus F) = minusnabla119901 + 120583nabla2u (1)

where 120588 is the fluid density F is a specific prescribed bodyforce (ie gravity force) and 120583 is the fluid dynamic viscosity

As in hydraulic flows at least two phases are alwayspresent water and air the above system of equations must beapplied to a multiphase systemThe simulation of multiphaseflows is a challenging task due to the inherent complexityof the involved phenomena (ie moving interfaces withcomplex topology) and represents one of the leading edgesof computational physics

The following two main approaches are widely used tosimulate multiphase flows the Euler-Lagrange approach andthe Euler-Euler approachThe latter approach usually chosenin hydraulics applications is based on the assumption thatthe volume of a phase cannot be occupied by other phasesthus introducing the concept of phase volume fractions ascontinuous functions of space and time

For the present work we employed the so-called volumeof fluid (VOF) method [31 32] designed for two or moreimmiscible fluids where only one fluid (ie air) is compress-ible and the position of the interface between the fluids isof interest The VOF method is a surface-tracking techniqueapplied to a fixed Eulerian mesh in which a specie transportequation is used to determine the relative volume fraction ofthe two phases or phase fraction in each computational cellPractically a single set of Reynolds-averaged Navier-Stokesequations is solved and shared by the fluids and for theadditional phase its volume fraction 120574 is tracked throughoutthe domain

Therefore the full set of governing equations for the fluidflow are as follows

nabla sdot u = 0

120597120588u120597119905+ nabla sdot (120588uu) minus nabla sdot ((120583 + 120583

119905) S) = minusnabla119901 + 120588g + 120590119870 nabla1205741003816100381610038161003816nabla120574

1003816100381610038161003816

120597120574

120597119905+ nabla sdot (u120574) = 0

(2)

where u is the velocity vector field 119901 is the pressure field 120583119905is

the turbulent eddy viscosity S is the strain rate tensor definedby S = (nablau + nablau119879)2 120590 is the surface tension and 119870 is thesurface curvature

The nature of the VOF method means that an interfacebetween the species is not explicitly computed but ratheremerges as a property of the phase fraction field Since thephase fraction can have any value between 0 and 1 the inter-face is never sharply defined but occupies a volume aroundthe region where a sharp interface should exist

Mathematical Problems in Engineering 3

Table 1 Velocity vectors for the D3Q19 models

ci Direction Number of cells 119891eq0119894

0 0 1 119891eq01

c 1 6 6 119891eq02

119888 sdot radic2 7 18 12 119891eq03

The model used in this work is based on an open sourcecomputational fluid dynamics (CFD) platform namedOpen-FOAM [33] freely available on the Internet OpenFOAMprimarily designed for problems in continuum mechanicsuses the tensorial approach and object-oriented techniquesIt provides a fundamental platform to write C++ new solversfor different problems as long as the problem can be writtenin tensorial partial differential equation form

22 The LBMNumerical Scheme This section introduces thetheoretical background of the proposed LBM with specificregard to the following

(i) the single-phase LBM and the boundary conditions

(ii) the front tracking concept

that are respectively inserted in the two following sections

221 Single-Phase Lattice Boltzmann Method and BoundaryConditions Artificial and natural fluvial hydrodynamicsmaybe simulated at the macroscopic scale using the LBM that isstructured as a mesoscopic (ie scaled larger than micro andsmaller than macro) method LBM treats the water mass byanalysing its density probability function 119891

119894(x ci 119905) where

x is the particle spatial location within the lattice domain119905 is the time variable and ci is the kinematic componentThe dynamic of the particle distribution function is describedin (3) that represents a discrete lattice Boltzmann equationrepresenting the evolution in time of the fluid flow byproviding a numerical solution of the modification of thedensity probability function

119891119894(x + ciΔ119905 119905 + Δ119905) minus 119891119894 (x 119905)

+ Δ119905 sdot Fi = 119862119894 (x 119905) 119894 = 1 119887(3)

where the left and right terms represent respectively themolecular free-streaming and the particle collisions Fi is anadditional term accounting for internal (eg collision) andexternal forcing processes (eg electric and magnetic fieldsor gravity)

For water-driven processes it is reasonable to assume adiscrete set of directions alongwhich particlesmay route andas a consequence ci is restricted to a maximum of 7 to 9 for2D or 19 directions for 3D flowsThis 19-direction discretizedcubic lattice model implemented here (namely D3Q19) ischaracterized by the set of predefined velocity vectors sche-matically represented in Figure 1 and reported in Table 1

c15

c12

c3

c9

c13

c18

c7

c0c8

c16c5

c4

c14

c10

c6

c2

c1

c14

c17

c11

Figure 1 D3Q19 velocity model

The collision component is integrated by implementingthe Bhatnagar-Gross-Krook Model (BGK) [34]

119862119894(x 119905) = minusΔ119905

120591[119891119894(x 119905) minus 119891eq

119894(x 119905)] (4)

that characterizes the density modifications caused by par-ticle collisions as related to dynamic modifications from thelocal equilibrium 119891eq

119894[35 36]

If the relative Knudsen and Mach numbers are smallenough this formulation is able to represent the dynamicsof a fluid with pressure 119875 = 1198882

119904120588 and a kinematic viscosity

120592 = 1198882119904(120591 minus Δ1199052) where 119888

119904is the lattice sound velocity [12]

The proposed numerical approach is implemented on astructured Cartesian lattice (ie mesh) where the particlesmay only reside on the mesh nodes

In the proposed model boundary nodes are solved usingthe bounce-back scheme meaning that particles collidingwith a stationary wall simply reverse their momentum [1237] In particular the half-way bounce-back scheme is hereapplied by posing the reflecting wall at a medium distancebetween boundary nodes and the physical walls [38] As aresult the proposed model can simulate both solid features(eg walls levees etc) of known geometry or open bound-aries of predefined velocity discharge or pressure function

The turbulence flow field is implemented in the pro-posed LBM model by employing a Large Eddy Simulation(LES) approach [39] that characterizes the flow field sim-ulating large-scale structures (resolved grid scales) whichare assumed to be mostly predominating in defining thetransport of mass and momentum and not directly sim-ulating the small-scale local effects The definition ofthe boundary between small and large processes is based onthe Smagorinsky model [40]


222 Front-Tracking Model In the Front-Tracking (FT) for-mulation proposed here [23] the interface is treated as a zerothickness mathematical surface across which the densityfield jumps from the light to the dense phase and viceversa and is tracked throughout the computational domainthrough an additional scalar variable (ie liquid mass frac-tion) A single set of LB equations is solved for both the heavyfluids with respect to the light fluids (ie gas) in agreementwith the front-tracking methodology that do not aim at pro-viding any description of the physics of the phase-transitionbetween the two phases The main difference which alsorepresents the main advantage of the LBM front-trackingapproachwith respect to the traditional continuummethodsis that the single-species transport equation used in Navier-StokesVOF model to determine the relative volume fractionof the two phases in each computational cell is not neededsince the free-surface tracking is automatically performedby advancing the fluid elements In practical terms onlythe liquid phase flow is numerically solved while the gascharacterization is neglected and an additional variable 120576(ie the liquid volume fraction) is structured to track the freesurface throughout the computational domain This variableis numerically solved using the following equation

120576 (x 119905) = 119898 (x 119905)120588 (x 119905)

120576 (x) = 0 forallx isin 119866120576 (x) = 1 forallx isin 1198710 lt 120576 (x) lt 1 forallx isin 119868

(5)

inwhich119866 119871 and 119868 represent respectively empty (gas) liquidand interface cells

The computation scheme is integrated numerically inthree steps as follows (1) interface positioning and flowmassmotion update defined by analysing cell fluxes (2) definitionof the boundary conditions at the interface to separate the gasand liquid state (3) update of cell type As a result interfacecells defined by the x position and time 119905 are defined asfollows

Δ119898119894=

0 forallx isin 119866119891pc119894(x + ciΔ119905 119905) minus 119891

pcminus119894(x 119905) forallx isin 119871

(120576 (x 119905) + 120576 (x + ciΔ119905 119905)

2)

times [119891pc119894(x + ciΔ119905 119905) minus 119891

pcminus119894(x 119905)] forallx isin 119868

(6)

in which pc characterizes the postcollision state and the signis related to the mass direction where minus119894 has opposite direc-tion with respect to 119894 Mass dynamics in space and time aredefined as

119898(x 119905 + Δ119905) = 119898 (x 119905) +18

sum119894=1

Δ119898119894(x 119905) (7)

Combining (6) and (7) the continuity principium is inrespect with the condition that direct modification of statefrom liquid to gas or vice versa is not permitted outside theinterface In fact liquid and gas cells are only allowed totransform into interface cells whereas interface cells can be

0

Removeddam wall

1 2 3

1198670

ℎ

119909min 119909 119909max1199090

Figure 2 Schematic representation of the water surface profile fora 20m long straight channel dam break

transformed into both gas and liquid cells In this way massand density are completely decoupled and as a results massevolution does not affect the particle distribution functions

3 Numerical Results for Three Case Studies

Theproposed LBM andNS are applied for the following threetest cases

(a) the free-surface flow routed along a straight channelwith LBM modeling results that are validated bycomparison with two different types of analyticalsolutions

(b) the free-surface flow routed along a straight channelderiving from the partial asymmetrical collapse ofa vertical dam embankment with results that arecompared in terms of hydraulics and computationalefficiency

(c) the simulation of a severe transient flow through asynthetic urban district

The three tests are hereafter presented in detail

31 Test (a) Fast Flow Propagating along a Straight ChannelThe routing of a fast moving flow resulting from a dambreak propagated through a 20m long straight channel(see Figure 2) is simulated and the numerical results arecompared to the analytical solution provided by Ritter [41]

The channel bed is horizontal Bottom and wall frictionis set to null The numerical mesh is made up of 2000 cubicelements with a grid-spacing of 001m At time zero a 05mby 7m volume of water ideally positioned at the mostupstream section of the channel is released instantaneouslyand two resulting waves propagate respectively downstreamand upstream back to the reservoir At a given time 119905 from thestart of the simulation (ie dambreakbreach or ideal verticalwall removal) three different regions are defined (Figure 2) asfollows

(1) for 119909 lt 119909min we have ℎ = 1198670 = 05m


0 2 4 6 8 10 12 14 16 18 20[119909 (m)]

0

01

02

03

04

05

[ℎ(m

)]

Chanson [43]Ritter [41]LBM model

Figure 3 LBM simulation results analytical solution and experi-mental data for the 20m long straight channel dam break at times05 s 1 s 2 s and 3 s

(2) for 119909min lt 119909 lt 119909max the water surface profile isparabolic

ℎ =1198670

(119909min minus 119909max)2(1199092minus 2119909max119909 + 119909max

2)

119909min = minus119905radic1198921198670 + 1199090

119909max = 2119905radic1198921198670 + 1199090

(8)

(3) for 119909 gt 119909max we have ℎ = 0

where ℎ is the water elevation Figure 3 shows the agreementbetween numerical and analytical data apart some minorexpected differences at the boundaries [42] In fact whereRitterrsquos analytical scheme is not suited for this comparisonother methods were selected like the Chansonrsquos solution [43]

8

3

1

119891

(1 minus 1198802)

1198802

3

=119905

radic1198921198670

119909119904

1198670

=1199090

1198670

+ (3

2119880 minus 1)

119905

radic1198921198670

+4

1198911198802(1 minus

119880

2)4

ℎ

1198670

=1

9(2 minus

(119909 minus 1199090)

1198670(119905radic119892119867

0))

2

if minus 119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le (3

2119880 minus 1)

119905

radic1198921198670

ℎ

1198670

= radic119891

41198802(119909119904minus 119909)

1198670

if (32119880 minus 1)

119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le(119909119904minus 1199090)

1198670

(9)

that is also plotted in Figure 3

minus18 minus14 minus10 minus6 minus2 2 6 10 14 18 220

005

01

015

02

025

03

119905 = 0 s

[119909 (m)]

[ℎ(m

)]

119889(023m)

(a)


010203040506070809

111

[ℎ119889

]

119905 = 568 s

Experimental [44] LBM modelChanson [43]

Ritter [41]

[119909(119889 lowast 119905)]

(b)

Figure 4 Comparison between numerical LBM and analytical andexperimental results at time 568 s for the 40m long straight channeldam break

To further prove the validity of the code a dam breakflow propagating along a 40m long straight channel issimulated At time zero a 023m by 18m water volume isreleased instantaneously In Figure 4 LBM numerical resultsare compared to the Ritter [41] and Chanson [43] as well asto the experimental measurements provided by [44] Thecomparison is performed in terms of dimensionless instan-taneous free-surface profiles (ℎ119889) between numerical andanalytical data (assuming 119891 = 002)

32 Test (b) Asymmetrical Dam BreakmdashLBM and NS Numer-ical Models The second test is characterized by the simu-lation of a submersion wave caused by the partial collapseof a nonsymmetrical sluice gate (or partial asymmetric dambreach) The spatial 3D computational domain is 200m times

200m times 20m where the boundaries of the domain areimpermeable walls The nonuniform initial condition for theliquid phase fraction is specified the water surface levelis initially 10m for the upstream section and 5m for thedownstream section An unsteady flow is generated by


00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA

Figure 5 Geometric schematization in plan view for test (b)Sections A-A and B-B and Points P1 (100 130) P2 (110 130) P3 (130130) and P4 (150 130) are shown All dimensions are in m

the instantaneous collapse of an asymmetrical 75m longportion of the dam (see Figure 5) The bottom is flat andground resistance to the motion is neglected

Numerical results of several authors are available in theliterature for this case [6 45 46] that is a standard bench-mark for hydraulic models in dam break applications Wechose to compare the LBM and NS described in the previoussection in terms of computational efficiency Calculationshave been performed with a time step of 001 s on a com-putational mesh of 800000 nodes (ie lattice sites for theLBM) Figure 6 shows the comparison of the computed waterlevel 72 seconds after the breach when the flow reaches theleft side of the tank while Figure 7 shows the simulated watersurface levels along sections A-A (longitudinal 119884 = 130m)and B-B (transverse 119883 = 110m) Water level hydrographs atselected points of the domain have been also evaluated andFigure 8 shows the results obtained for points P1 and P3 withboth LBM and NS approaches

Results show that NS and LBM using the same gridresolution are in good agreement in terms of water heightsand hydrograph along the section AA while the results forthe section BB (Figure 7(b)) are remarkably different espe-cially in terms of water depths This is expected with specificregard to the central part of the domain where the dam breakoriginates and the flow is significantly 3D while in the otherportions of the domain through which the emptying of thesystem applies (from the noncollapsing area towards wherethe breach is) in the direction of the flow that is parallel to thegate the two models give different results with greater waterdepths evaluated by NS

From the computational point of view using the samehardware the LBMmodel has completed the tests in one-fifth

Table 2 Computational efficiency of LBM and NS

MathematicalModel Δ119909

Computationaltime (s)

Simulationtime (s)

NS 10 6941 72LBM 10 1512 72LBM 08 301104 72LBM 06 46764 72LBM 04 6350 72LBM 03 730008 72

of the time needed by the NS Results are summarized inTable 2 that also shows that the LBM code is able to providea higher level of details of the flow pattern using the samecomputational time or to provide the same accuracy anddetails in less time Figure 9 synthesizing the LBM versusNS computational efficiency comparison test shows thatthe LBM model can use a mesh 65 denser than NS one(Δ119883(LBM) = 035 cm Δ119883(NS) = 1m) for the samecomputational time

33 Test (c) Severe Transient Flow through a Synthetic UrbanDistrict This test presents the simulation of the effects ofa severe transient flow on a synthetic urban area This casestudy replicates the experiment developed at the HydraulicLaboratory of the Civil and Environmental EngineeringDepartment of the Universite Catholique de Louvain inBelgium [47] The geometric configuration is characterizedby a regular distribution of rectangular obstacles (buildings)intercepting the flow in geometric domain The realisticaverage ratio between building and street widths of 1 to 3was selected by interpreting a real urban configuration (theaerial view of Brussels in Belgium) Buildingsrsquo heights aresimulated as high enough in order not to be overtopped by theflow The experimental physical setup consists of a 36m longby 36m wide channel with a flat profile (Figure 10) Build-ings wooden parallelepipeds of 30 cm by 30 cm and streetsthat are 10 cm wide are positioned along the flow directionThe initial reservoir water level is 040m upstream Watersurface measurements were performed [47] by means ofseveral resistive level gauges for estimating the water surfacegeometric and dynamic properties (depths and surface veloc-ity fields)

The aim of this test is to investigate the potential of LBMand NS to correctly represent the complex dynamics of tran-sient flows In particular the physics of interest is related tothose flood phenomena propagated through urban settingfor which an initial severe transient phase is often followedby a slow transient that could approximate under certainsteady state conditions Nevertheless since the first instantscorrespond to a severe transient flow involving specific flowfeatures characterized by high velocities and transcriticalflows wave impacts on structures or higherwater rise follow-ing wave reflection against a structure the LBM and NS arechallenged to provide such unsteady irregular behaviorfollowed by the regular steady conditions of the followingsequences


00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84

3D Navier-Stokes model

(b)

Figure 6 Comparison of the computed water level 72 seconds after the breachmdashLBM versus and NS are respectively plotted in the left andright

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)

Figure 7 Comparison between water surface elevations obtained with LBM and NS in the A-A and B-B sections at 72 seconds after thefailure


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)

Figure 8 Water level hydrographs at points P1 (100 130) and P3 (130 130) for test (b) Continuous line LBM scatter points for NS

0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)

Figure 9 Computational comparison test for and LBMmodels for test (b)

331 Simulation SetupmdashMesoscopic Approach Using LBMDue to the large-scale of the domain in horizontal119883119884 planethe test case has been replicated using a computational meshof cubes of 0025m resolution producing a total number of4976640 nodes The downstream region has been repre-sented assigning a liquid volume fraction of 05 to the firstvertical row of nodes Considering the dimension of thedomain the mesh is quite rough and only few computationalnodes describe the streets that are only 01m wide

332 Simulation SetupmdashContinuous ApproachUsingNS Forall simulations the computational domain is the same withthe following dimensions 36m long 36m wide and 1mhigh Five scenarios have been developed for the NS simu-lation varying the resolution of the mesh in the three direc-tionsThe computationalmesh for each scenario derives from

the intersection and refinement of the mesh composed ofhexahedra (configuration A see Table 3) with the 3D surfaceof the synthetic urban setting (Figure 11) In configuration Aa grid variation along the vertical with a ratio of 1 to 10 isused The result of such a refinement process is representedby configuration B (see Table 3) For the five scenarios M1 toM5 the diverse grid resolutions lead to discrepancies betweenthe numerical results and physical measurements both insideand outside the city The comparison between experimentaldata and numerical results provided by the LBM and the NSscenarios M1 M2 M5 for the section 119910 = 2m at 119905 = 5 6 and10 s after the break are presented in Figure 12

The free-surface profile includes a hydraulic jump at 5 safter the break In the graphs for 119905 = 5 s and 119905 = 6 s when theflow depth in the city is still low the flow reflection against thebuildings is clearly visible for both LBM and NS simulations


Table 3 Geometric characteristics of computational meshes adopted for scenarios M1 M2 M3 M4 and M5

M1 M2 M3 M4 M5Number of TOT cells 324119864 + 04 259119864 + 05 518119864 + 05 104119864 + 06 207119864 + 06

Min volume cell (m3) 979119864 minus 04 125119864 minus 04 633119864 minus 05 316119864 minus 05 158119864 minus 05

Max volume cell (m3) 979119864 minus 03 125119864 minus 03 633119864 minus 04 316119864 minus 04 158119864 minus 04

Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06

(180 times 18 times 10) (360 times 36 times 20) (360 times 36 times 40) (360 times 72 times 40) (720 times 72 times 40)Number of faces 102119864 + 05 798119864 + 05 158119864 + 06 315119864 + 06 630119864 + 06

Number of TOT cells 616119864 + 04 401119864 + 05 796119864 + 05 112119864 + 06 202119864 + 06

Min volume cell 610119864 minus 06 150119864 minus 06 770119864 minus 07 310119864 minus 06 125119864 minus 05

Max volume cell 979119864 minus 03 125119864 minus 03 633119864 minus 04 316119864 minus 04 158119864 minus 04

Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13

Figure 10 Schematic representation of experimental and computational domain

7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)

Figure 11 Computational mesh definition procedure Example of intersection between 3D obstacle surface and original mesh


115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)

Section 119884 = 2 m

(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s

LBM modelNS model (M5)NS model (M2)

NS model (M1)Experimental

Section 119884 =

119883 (m)

2 m

(c)

Figure 12 Comparison between experimental and simulated free-surface value for the section 119884 = 2m for scenarios M1 M5 of continuousapproach and LBM for 119905 = 5 6 and 10 s after the break

and the LBMmodel correctly predicts the free surface at low119909 values as the NS model with the finest mesh At 119905 = 10 sthe upstream hydraulic jump is still present but the waterlevel has significantly increased in the streets The flow inthe streets evolves from supercritical Fr gt 1 (with a controlsection near the street entrance) to subcritical (with a controlsection at the street exit)

The overall agreement between the numerical and exper-imental results is promising for the use of the mesoscopicnumerical model for replicating the complex physics ofrapidly varying flows in such a dense urban area even if dueto the mesh resolution results of lattice-based simulationsare less accurate than the NS because of the mesh refinement(especially for M5) Analyzing the effect of mesh refinementthe obstacle mesh refinement of M3ndashM5 is able to betterreproduce the unsteady hydraulic features and flow processeslike hydraulics jumps while the coarser meshes of M1-M2are characterized by some errors where the flow behavior isirregular

For LBM at the time steps 119905 = 5 and 6 s significant differ-ences appear between the two numerical models For exam-ple the depression and the hydraulic jump at the entranceof the urban district (119909 = 520m) is absent in the LBMcoarse mesh result One limitation of the used LBM code isthat the computationalmesh is not refined locally for examplein a single area of interest This disadvantage with respect to

the used NS model clearly indicates the utility of employingmesh refinement techniques as those developed for standardLBM [48]

Figure 13 shows the results at 119905 = 4 seconds for 5 differentmeshes while in Figure 14 snapshots of free-surface evolutionat different time steps (0 2 4 6 8 and 10 s) are insertedFigure 15 reports the value of velocitymagnitude for the sametime steps

Observations indicated that the flow rises at the city frontbefore entering the streets after the wave impact a phe-nomenon that is similar to the impact against a single obstaclerather than a series of obstacles A hydraulic jump is repre-sented at the impact section (Figure 14) with the water levelthat is locally higher with respect to the same case withoutbuildings while a wake zone is developed immediatelydownstream of the city

4 Conclusions

In this study an experimental fully 3D flow modeling frame-work is implemented to numerically simulate the unsteadyirregular flows originating from dam breaks with specificregard to the complex interaction between the geometricalurban features and the associated physical hydraulic pro-cesses The proposed novel technique is based on a front-tracking variant of the lattice Boltzmann method that is here


12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)

Figure 13 Comparison of experimental free-surface value (points) and simulated fraction fill results for 6 different computational meshes attime = 4 s from the release at section 119884 = 2m

investigated as a valid alternative to the standard numericalNavier-Stokes model in the treatment of instantaneouslyvarying flows interacting with artificial structures In partic-ular test cases evaluate the accuracy and the computationalefficiency of numerical simulations with respect to analyt-ical solutions and experimental data The critical analysisof model performances provides the following conclusiveremarks

The proposed LBM is able to predict with a high level ofaccuracy typical hydraulic phenomena characterizing engi-neering applications related to the flooding of simple straight

channels (test (a)) and asymmetrical front waves in rectan-gular basins (test (b)) More specifically as compared to thefully 3D NS model it is reported the validity and robustnessof LBM results that while implementing the same resolutionof the geometric domain are demonstrated to be morecomputationally efficient (test (b))

The last test investigates the diverse hydraulic unsteadyprocesses like the hydraulic jump return and diverged flowthat originate from the passage of the flood wave througha synthetic urban setting (test (c)) The application of thepresented 3Dnumerical schemes shows themajor advantages


04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)

Figure 14 Free-surface evolution for scenario M5 for different time steps ((a) = 0 s (b) =2 s (c) = 4 s (d) = 6 s (e) = 8 s and (f) =10 s)

00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)

Figure 15 Evolution of velocity magnitude for scenario M5 atdifferent time stepsmdash119905 = 4 s (a) and 6 s (b)

and disadvantages of both LBM and NS schemes confirmingthe ability of LBM in representing such abruptly changing andirregular flow dynamics Nevertheless the mesoscopic LBMapproach is significantly bound by the regular structuredmesh while the continuous NS model performs well also inlarge domains especially when using a more accurate grid

(M5) that is adaptively refined to more accurately representobstacles

References

[1] P Manciola S Di Francesco and C Biscarini ldquoFlood protec-tion and risk management the case of Tescio River basinrdquoin Proceedings of the Role of Hydrology in Water ResourcesManagement Symposium vol 327 of IAHS Red Book Series pp174ndash183 Capri Italy October 2008

[2] Y Jia and S S Y Wang ldquoNumerical model for channel flowand morphological change studiesrdquo Journal of Hydraulic Engi-neering vol 125 no 9 pp 924ndash933 1999

[3] P G Manciola A Mazzoni and F Savi ldquoFormation and prop-agation of steep waves an investigative experimental interpre-tationrdquo in Proceedings of the Specialty Conference on Modellingof Flood Propagation Over Initially Dry Areas pp 283ndash297Milano Italy July 1994

[4] C Biscarini ldquoComputational fluid dynamics modeling of land-slide generated water wavesrdquo Landslide Springer vol 7 no 2 pp117ndash124 2010


[5] A Ferrari L Fraccarollo M Dumbser E F Toro and AArmanini ldquoThree-dimensional flow evolution after a dambreakrdquo Journal of Fluid Mechanics vol 663 pp 456ndash477 2010

[6] C Biscarini S Di Francesco and P Manciola ldquoCFDmodellingapproach for dam break flow studiesrdquo Hydrology and EarthSystem Sciences vol 14 no 4 pp 705ndash718 2010

[7] D Dutykh and D Mitsotakis ldquoOn the relevance of the dambreak problem in the context of nonlinear shallow waterequationsrdquo Discrete and Continuous Dynamical Systems B vol13 no 4 pp 799ndash818 2010

[8] H Liu J G Zhou and R Burrows ldquoLattice Boltzmann simula-tions of the transient shallow water flowsrdquo Advances in WaterResources vol 33 no 4 pp 387ndash396 2010

[9] C L Mader ldquoModeling the 1958 Lituya Bay Mega-Tsunami IIrdquoScience of Tsunami Hazards vol 20 pp 241ndash250 2002

[10] C Biscarini andM Testa ldquoThree-dimensional numerical mod-elling of the marmore water fallsrdquo Progress in ComputationalFluid Dynamics vol 11 no 22011 pp 105ndash115 2011

[11] R Benzi S Succi and M Vergassola ldquoThe lattice Boltzmannequation theory and applicationsrdquo Physics Report vol 222 no3 pp 145ndash197 1992

[12] S SucciTheLattice BoltzmannEquation for FluidDynamics andBeyond Numerical Mathematics and Scientific ComputationThe Clarendon Press Oxford UK 2001

[13] S Ubertini P Asinari and S Succi ldquoThree ways to latticeBoltzmann a unified time-marching picturerdquo Physical ReviewE vol 81 no 1 Article ID 016311 2010

[14] J F Chau D Or andM C Sukop ldquoSimulation of gaseous diffu-sion in partially saturated porous media under variable gravitywith lattice BoltzmannmethodsrdquoWater Resources Research vol41 no 8 Article IDW08410 2005

[15] X Shan and H Chen ldquoLattice Boltzmannmodel for simulatingflows withmultiple phases and componentsrdquo Physical Review Evol 47 pp 1815ndash1819 1993

[16] D Chiappini G Bella S Succi F Toschi and S UbertinildquoImproved lattice Boltzmann without parasitic currents forRayleigh-Taylor instabilityrdquo Communications in ComputationalPhysics vol 7 no 3 pp 423ndash444 2010

[17] G Falcucci S Ubertini and S Succi ldquoLattice Boltzmannsimulations of phase-separating flows at large density ratios thecase of doubly-attractive pseudo-potentialsrdquo Soft Matter vol 6no 18 pp 4357ndash4365 2010

[18] Y Peng J G Zhou and R Burrows ldquoModeling free-surfaceflow in rectangular shallow basins by using lattice boltzmannmethodrdquo Journal of Hydraulic Engineering vol 137 no 12 pp1680ndash1685 2012

[19] G Falcucci G Bella G Chiatti S ChibbaroM Sbragaglia andS Succi ldquoLattice Boltzmann models with mid-range interac-tionsrdquo Communications in Computational Physics vol 2 no 6pp 1071ndash1084 2007

[20] C Biscarini S Di Francesco F Nardi and P Manciola ldquoThree-dimensional hydraulicmodelling of unsteady free-surface flowsusing the Lattice Boltzmann Methodrdquo Journal of HydraulicEngineering In review

[21] J G Zhou Lattice BoltzmannMethods For ShallowWater flowsSpringer 2004

[22] Y Li and P Huang ldquoA coupled lattice Boltzmann model for theshallow water-contamination systemrdquo International Journal forNumerical Methods in Fluids vol 59 pp 195ndash213 2009

[23] C Korner M Thies T Hofmann N Thurey and U RudeldquoLattice Boltzmann model for free surface flow for modeling

foamingrdquo Journal of Statistical Physics vol 121 no 1-2 pp 179ndash196 2005

[24] NThurey T Pohl U Rude M Ochsner and C Korner ldquoOpti-mization and stabilization of LBM free surface flow simulationsusing adaptive parameterizationrdquoComputers and Fluids vol 35no 8-9 pp 934ndash939 2006

[25] C Biscarini S D Francesco and M Mencattini ldquoApplicationof the lattice Boltzmann method for large-scale hydraulicproblemsrdquo International Journal of Numerical Methods for Heatand Fluid Flow vol 21 no 5 pp 584ndash601 2011

[26] G Falcucci S Ubertini C Biscarini et al ldquoLattice boltzmannmethods for multiphase flow simulations across scalesrdquo Com-munications in Computational Physics vol 9 no 2 pp 269ndash2962011

[27] H M Solo-Gabriele and F E Perkins ldquoStreamflow and sus-pended sediment transport in an urban environmentrdquo Journalof Hydraulic Engineering vol 123 no 9 pp 807ndash811 1997

[28] S Soares-Frazao and Y Zech ldquoExperimental study of dam-break flow against an isolated obstaclerdquo Journal of HydraulicResearch vol 45 pp 27ndash36 2007

[29] S Di Francesco C Biscarini and P Manciola ldquoMesoscopicmodeling of dam break flowrdquo in Proceedings of the 3rd Interna-tional Junior Researcher and Engineer Workshop on HydraulicStructures (IJREWHS rsquo10) R Jannsenn and H Chanson EdsHydraulic Model Report CH8010 School of Civil Eng TheUniversity of Queensland Brisbane Australia 2010

[30] D Yu and S N Lane ldquoUrban fluvial flood modelling using atwo dimensional diffusion-wave treatment 1 Mesh resolutioneffectsrdquo Hydrological Processes vol 20 pp 1541ndash1565 2006

[31] S Zaleski J Li and S Succi ldquoTwo-dimensional Navier-Stokessimulation of deformation and breakup of liquid patchesrdquoPhysical Review Letters vol 75 pp 244ndash247 1995

[32] O Ubbink and R I Issa ldquoA method for capturing sharpfluid interfaces on arbitrary meshesrdquo Journal of ComputationalPhysics vol 153 no 1 pp 26ndash50 1999

[33] Open FOAM User Guide Version 15 The Open Source CFDToolbox 2008

[34] P L Bhatnagar E P Gross andMKrook ldquoAmodel for collisionprocesses in gases I Small amplitude processes in charged andneutral one-component systemsrdquo Physical Review vol 94 no3 pp 511ndash525 1954

[35] S Chen and G D Doolen ldquoLattice Boltzmannmethod for fluidflowsrdquo Annual Review of Fluid Mechanics vol 30 pp 329ndash3641998

[36] F Tosi S Ubertini S Succi H Chen and I V KarlinldquoNumerical stability of entropic versus positivity-enforcinglattice Boltzmann schemesrdquo Mathematics and Computers inSimulation vol 72 no 2-6 pp 227ndash231 2006

[37] P A Skordos ldquoInitial and boundary conditions for the latticeBoltzmannmethodrdquo Physical Review E vol 48 no 6 pp 4823ndash4842 1993

[38] T Inamuro M Yoshino and F Ogino ldquoA non-slip boundarycondition for lattice Boltzmann methodrdquo Journal of Computa-tional Physics vol 155 p 307 1999

[39] B Galperin and S A Orszag Large Eddy Simulation of Com-plex Engineering and Geophysical Flows Cambridge UniversityPress New York NY USA 1993

[40] J Smagorinsky ldquoGeneral circulation experiments with theprimitive equations I The basic experimentrdquoMonthly WeatherReview vol 91 pp 99ndash164 1963


[41] A Ritter ldquoThe propagation of water wavesrdquo Ver Deutschingenieur zeitschr vol 36 no 33 part 3 pp 947ndash954 1892

[42] R Dressler ldquoComparison of theories and experiments for thehydraulic dam-break waverdquo Proceedings of the InternationalAssociation of Scientific Hydrology Assemblee Generale RomeItaly vol 3 no 38 pp 319ndash328 1954

[43] H Chanson ldquoApplication of themethod of characteristics to thedambreakwave problemrdquo Journal of Hydraulic Research vol 47no 1 pp 41ndash49 2009

[44] Y Cavaille Contribution a lrsquoetude de lrsquoecoulement variableaccompagnant la vidange brusque drsquoune retenue vol 410 ofPublications Scientifiques et Techniques du Ministere de lrsquoairParis France 1965

[45] F Alcrudo and P Garcia-Navarro ldquoHigh-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equa-tionsrdquo International Journal for Numerical Methods in Fluidsvol 16 no 6 pp 489ndash505 1993

[46] R J Fennema and M H Chaudhry ldquoExplicit methods for 2-Dtransient free-surface flowsrdquo Journal of Hydraulic Engineeringvol 116 no 8 pp 1013ndash1034 1990

[47] S Soares-Frazao and Y Zech ldquoDam-break flow through anidealised cityrdquo Journal of Hydraulic Research vol 46 no 5 pp648ndash658 2008

[48] O Filippova and D Hanel ldquoGrid Refinement for Lattice-BGKModelsrdquo Journal of Computational Physics vol 147 no 1 pp219ndash228 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


approach that also preserves the conservation law [12 13]LBM has been adopted for the characterization of flows inporous media [14] and for multiphase flows [15ndash17] amongothers [18 19] The LBM demonstrated to work efficientlyfor describing the complex physics of non ideal flows andcomplex geometries [20] There are also several works inves-tigating the performances of the LBM in simulating shallowfree-surface flows [21 22] but with no emphasis on their usefor practical engineering applications

In this work a fully 3D front-tracking formulation of theLBM that was originally presented [23] for foam dynamicsand that was revisited for free-surface flow applications [24ndash26] is implemented validated and compared to the standardNS approach for transient hydraulic flows including a syn-thetic case of flow propagation through an urban setting

The storage effect of urban areas as well as the hydrographseparation effect resulting in different water transmissionrates was investigated in several works [27] Experiments ofdam breaks were conducted [28] to investigate the effect ofa flood flow propagating to inundate a single building [29]Nevertheless the application and potential validation of LBMfor urban modeling of flash floods [30] has not been ade-quately developed yet

The novel aspect of the proposedwork is the investigationfor the potential of the LBM to simulate the flooding in highlyurbanized areas with specific regard to the detailed char-acterization of instantaneous discharge variations like theone caused by dam or dyke breaks or by intense rainfall offlash floods producing multidimensional street surface flowpaths which is one of the most complex and difficulthazardous phenomena to manage prevent and control Theinteraction of water flow with anthropogenic features pro-duces complex 3D flow structures that shall be simulated bytaking into consideration all the resulting flow physics suchas hydraulics jumps flow constrictions and back pressureeffects Disregarding those geometric and hydraulic featureswould result in a misinterpretation of the estimated waterlevels providing an inaccurate flood hazard and risk charac-terization

In this study given the impossibility of gathering mea-sured data on such phenomena a comparative analysis of theperformance of LBM and NS in simulating flash flood prop-agation in urban areas is investigated by a comparison withexperimental data of dam break hydraulic effects on asynthetic case study represented by an ideal city

The present document is organized as follows in the nextsection the theoretical and numerical specifications of bothLBM and NS are introduced and described Then in the casestudy section the results of the application of the proposedmethods are insertedwith specific regard to (1) the test case ofthe LBM results compared to an analytical solution (2) theanalysis and comparison of LBMandNS codes applied on thesimulation of a rapidly varying flow along a simple geometryof a straight river reach (3) the evaluation and validation ofthe LBM and NS code application on the flow propagatingthrough a synthetic urban domain by means of comparisonwith experimental data of a physical model The last twosections represent respectively the discussion and conclusiveremarks

2 Methods

21 The Navier-Stokes Numerical Scheme The most generalmacroscopic model for hydraulic flows may be representedby the incompressible Navier-Stokes equations as follows

120588(120597u120597119905minus F) = minusnabla119901 + 120583nabla2u (1)

where 120588 is the fluid density F is a specific prescribed bodyforce (ie gravity force) and 120583 is the fluid dynamic viscosity

As in hydraulic flows at least two phases are alwayspresent water and air the above system of equations must beapplied to a multiphase systemThe simulation of multiphaseflows is a challenging task due to the inherent complexityof the involved phenomena (ie moving interfaces withcomplex topology) and represents one of the leading edgesof computational physics

The following two main approaches are widely used tosimulate multiphase flows the Euler-Lagrange approach andthe Euler-Euler approachThe latter approach usually chosenin hydraulics applications is based on the assumption thatthe volume of a phase cannot be occupied by other phasesthus introducing the concept of phase volume fractions ascontinuous functions of space and time

For the present work we employed the so-called volumeof fluid (VOF) method [31 32] designed for two or moreimmiscible fluids where only one fluid (ie air) is compress-ible and the position of the interface between the fluids isof interest The VOF method is a surface-tracking techniqueapplied to a fixed Eulerian mesh in which a specie transportequation is used to determine the relative volume fraction ofthe two phases or phase fraction in each computational cellPractically a single set of Reynolds-averaged Navier-Stokesequations is solved and shared by the fluids and for theadditional phase its volume fraction 120574 is tracked throughoutthe domain

Therefore the full set of governing equations for the fluidflow are as follows

nabla sdot u = 0

120597120588u120597119905+ nabla sdot (120588uu) minus nabla sdot ((120583 + 120583

119905) S) = minusnabla119901 + 120588g + 120590119870 nabla1205741003816100381610038161003816nabla120574

1003816100381610038161003816

120597120574

120597119905+ nabla sdot (u120574) = 0

(2)

where u is the velocity vector field 119901 is the pressure field 120583119905is

the turbulent eddy viscosity S is the strain rate tensor definedby S = (nablau + nablau119879)2 120590 is the surface tension and 119870 is thesurface curvature

The nature of the VOF method means that an interfacebetween the species is not explicitly computed but ratheremerges as a property of the phase fraction field Since thephase fraction can have any value between 0 and 1 the inter-face is never sharply defined but occupies a volume aroundthe region where a sharp interface should exist




0 0 1 119891eq01

c 1 6 6 119891eq02

119888 sdot radic2 7 18 12 119891eq03







119894(x ci 119905) where


119891119894(x + ciΔ119905 119905 + Δ119905) minus 119891119894 (x 119905)

+ Δ119905 sdot Fi = 119862119894 (x 119905) 119894 = 1 119887(3)



c15

c12

c3

c9

c13

c18

c7

c0c8

c16c5

c4

c14

c10

c6

c2

c1

c14

c17

c11



119862119894(x 119905) = minusΔ119905

120591[119891119894(x 119905) minus 119891eq

119894(x 119905)] (4)


119894[35 36]



120592 = 1198882119904(120591 minus Δ1199052) where 119888







120576 (x 119905) = 119898 (x 119905)120588 (x 119905)


(5)



Δ119898119894=



(120576 (x 119905) + 120576 (x + ciΔ119905 119905)

2)



(6)


119898(x 119905 + Δ119905) = 119898 (x 119905) +18

sum119894=1

Δ119898119894(x 119905) (7)


0

Removeddam wall

1 2 3

1198670

ℎ

119909min 119909 119909max1199090













0 2 4 6 8 10 12 14 16 18 20[119909 (m)]

0

01

02

03

04

05

[ℎ(m

)]




ℎ =1198670


2)

119909min = minus119905radic1198921198670 + 1199090

119909max = 2119905radic1198921198670 + 1199090

(8)



8

3

1

119891

(1 minus 1198802)

1198802

3

=119905

radic1198921198670

119909119904

1198670

=1199090

1198670

+ (3

2119880 minus 1)

119905

radic1198921198670

+4

1198911198802(1 minus

119880

2)4

ℎ

1198670

=1

9(2 minus

(119909 minus 1199090)

1198670(119905radic119892119867

0))

2

if minus 119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le (3

2119880 minus 1)

119905

radic1198921198670

ℎ

1198670

= radic119891

41198802(119909119904minus 119909)

1198670

if (32119880 minus 1)

119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le(119909119904minus 1199090)

1198670

(9)



005

01

015

02

025

03

119905 = 0 s

[119909 (m)]

[ℎ(m

)]

119889(023m)

(a)


010203040506070809

111

[ℎ119889

]

119905 = 568 s


Ritter [41]

[119909(119889 lowast 119905)]

(b)






00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA









Simulationtime (s)






00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 0 1 119891eq01

c 1 6 6 119891eq02

119888 sdot radic2 7 18 12 119891eq03







119894(x ci 119905) where


119891119894(x + ciΔ119905 119905 + Δ119905) minus 119891119894 (x 119905)

+ Δ119905 sdot Fi = 119862119894 (x 119905) 119894 = 1 119887(3)



c15

c12

c3

c9

c13

c18

c7

c0c8

c16c5

c4

c14

c10

c6

c2

c1

c14

c17

c11



119862119894(x 119905) = minusΔ119905

120591[119891119894(x 119905) minus 119891eq

119894(x 119905)] (4)


119894[35 36]



120592 = 1198882119904(120591 minus Δ1199052) where 119888







120576 (x 119905) = 119898 (x 119905)120588 (x 119905)


(5)



Δ119898119894=



(120576 (x 119905) + 120576 (x + ciΔ119905 119905)

2)



(6)


119898(x 119905 + Δ119905) = 119898 (x 119905) +18

sum119894=1

Δ119898119894(x 119905) (7)


0

Removeddam wall

1 2 3

1198670

ℎ

119909min 119909 119909max1199090













0 2 4 6 8 10 12 14 16 18 20[119909 (m)]

0

01

02

03

04

05

[ℎ(m

)]




ℎ =1198670


2)

119909min = minus119905radic1198921198670 + 1199090

119909max = 2119905radic1198921198670 + 1199090

(8)



8

3

1

119891

(1 minus 1198802)

1198802

3

=119905

radic1198921198670

119909119904

1198670

=1199090

1198670

+ (3

2119880 minus 1)

119905

radic1198921198670

+4

1198911198802(1 minus

119880

2)4

ℎ

1198670

=1

9(2 minus

(119909 minus 1199090)

1198670(119905radic119892119867

0))

2

if minus 119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le (3

2119880 minus 1)

119905

radic1198921198670

ℎ

1198670

= radic119891

41198802(119909119904minus 119909)

1198670

if (32119880 minus 1)

119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le(119909119904minus 1199090)

1198670

(9)



005

01

015

02

025

03

119905 = 0 s

[119909 (m)]

[ℎ(m

)]

119889(023m)

(a)


010203040506070809

111

[ℎ119889

]

119905 = 568 s


Ritter [41]

[119909(119889 lowast 119905)]

(b)






00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA









Simulationtime (s)






00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120576 (x 119905) = 119898 (x 119905)120588 (x 119905)


(5)



Δ119898119894=



(120576 (x 119905) + 120576 (x + ciΔ119905 119905)

2)



(6)


119898(x 119905 + Δ119905) = 119898 (x 119905) +18

sum119894=1

Δ119898119894(x 119905) (7)


0

Removeddam wall

1 2 3

1198670

ℎ

119909min 119909 119909max1199090













0 2 4 6 8 10 12 14 16 18 20[119909 (m)]

0

01

02

03

04

05

[ℎ(m

)]




ℎ =1198670


2)

119909min = minus119905radic1198921198670 + 1199090

119909max = 2119905radic1198921198670 + 1199090

(8)



8

3

1

119891

(1 minus 1198802)

1198802

3

=119905

radic1198921198670

119909119904

1198670

=1199090

1198670

+ (3

2119880 minus 1)

119905

radic1198921198670

+4

1198911198802(1 minus

119880

2)4

ℎ

1198670

=1

9(2 minus

(119909 minus 1199090)

1198670(119905radic119892119867

0))

2

if minus 119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le (3

2119880 minus 1)

119905

radic1198921198670

ℎ

1198670

= radic119891

41198802(119909119904minus 119909)

1198670

if (32119880 minus 1)

119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le(119909119904minus 1199090)

1198670

(9)



005

01

015

02

025

03

119905 = 0 s

[119909 (m)]

[ℎ(m

)]

119889(023m)

(a)


010203040506070809

111

[ℎ119889

]

119905 = 568 s


Ritter [41]

[119909(119889 lowast 119905)]

(b)






00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA









Simulationtime (s)






00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 20[119909 (m)]

0

01

02

03

04

05

[ℎ(m

)]




ℎ =1198670


2)

119909min = minus119905radic1198921198670 + 1199090

119909max = 2119905radic1198921198670 + 1199090

(8)



8

3

1

119891

(1 minus 1198802)

1198802

3

=119905

radic1198921198670

119909119904

1198670

=1199090

1198670

+ (3

2119880 minus 1)

119905

radic1198921198670

+4

1198911198802(1 minus

119880

2)4

ℎ

1198670

=1

9(2 minus

(119909 minus 1199090)

1198670(119905radic119892119867

0))

2

if minus 119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le (3

2119880 minus 1)

119905

radic1198921198670

ℎ

1198670

= radic119891

41198802(119909119904minus 119909)

1198670

if (32119880 minus 1)

119905

radic1198921198670

le(119909 minus 119909

0)

1198670

le(119909119904minus 1199090)

1198670

(9)



005

01

015

02

025

03

119905 = 0 s

[119909 (m)]

[ℎ(m

)]

119889(023m)

(a)


010203040506070809

111

[ℎ119889

]

119905 = 568 s


Ritter [41]

[119909(119889 lowast 119905)]

(b)






00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA









Simulationtime (s)






00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

25

25

50

50

75

75

100

100

125

125

150

150

175

175

200

200

P1 P2 P3 P4

30

95

95

119884

119883

B

B

AA









Simulationtime (s)






00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

200

97 921

521

553

521521

636

708672

7787

78742

742

742708

578

672

885

636

672

814814

3D LBM model

119884

119883

(a)

00

20

20

40

40

60

60

80

80

100

100

120

120

140

140

160

160

180

180

200

A

200

97

921

578

578

527

636

553

636

672

778

778

742

742

742708

708

885

672

814

814

84

84


(b)


0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 A-A section

ℎ(m

)

NS modelLBM model

119883 (m)

(a)

0 25 50 75 100 125 150 175 2004

5

6

7

8

9

10 B-B section

ℎ(m

)

NS modelLBM model

119884 (m)

(b)



0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 155

55

6

65

7

75

8

85

9

119905 (s)

119867(m

)

00

50

50

100

100

150

150

200

200

P1P2

P3P4

A A

119884

119883

Point P1 (119883 = 100m 119884 = 130m)

NS modelLBM model

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905 (s)

5

55

6

65

7

75

8

85

9

119867(m

)

Point P3 (119883 = 130m 119884 = 130m)

NS modelLBM model

(b)


0 08 16 24 32 4 48 56 64 720

1000

2000

3000

4000

5000

6000

7000

8000

Simulation time (s)

Com

puta

tiona

l tim

e (s)

LBM

LBM

LBM

LBM

LBM

LBM(Δ119909 = 03 m)

(Δ119909 = 04 m)

(Δ119909 = 05 m)LBM(Δ119909 = 06 m)

(Δ119909 = 07 m)

(Δ119909 = 08 m)

(Δ119909 = 1 m)

LBM (Δ119909 = 035m)NS (Δ119909 = 1m)











Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Number of points 378119864 + 04 280119864 + 05 548119864 + 05 108119864 + 06 216119864 + 06





Number of points 835119864 + 04 484119864 + 05 937119864 + 05 121119864 + 06 212119864 + 06

Number of faces 207119864 + 05 129119864 + 06 253119864 + 06 344119864 + 06 617119864 + 06

675358

08

36

13

5

03 012

ℎ = 04m 1

13


7891011121314 6 5005115225335

15

(a)

7891011121314005115225335

(b)

7891011121314 6 515005115225335

119884

119883

(c)



115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 5 s

Section 119884 =

119883 (m)

2 m

(a)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 6 s

119883 (m)


(b)

115 12 125 13 135 14 145 150

01

02

03

04

05

ℎ(m

)

119905 = 10 s



Section 119884 =

119883 (m)

2 m

(c)








4 Conclusions



12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12 13 14 150

5

10

15

20

25

M1119867

(cm

)

119883 (m)

(a)

M2

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(b)

M3

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(c)

M4

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(d)

M5

12 13 14 150

5

10

15

20

25

119867(c

m)

119883 (m)

(e)







04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










04

20

(a)

04

20

(b)

04

20

(c)

04

20

(d)

20

04

(e)

04

20

(f)


00

36

36

0

5 10 15 20

0 5 10 15 200

04

08

12

16

2

Velo

city

mag

nitu

de (m

s)

(a)

(b)




References



























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Detailed Simulation of Complex Hydraulic Problems...

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Transcript of Research Article Detailed Simulation of Complex Hydraulic Problems...